Question

The points \((2,1)\) and \((-3,-4)\) are opposite vertices of a parallelogram. If the other two vertices lie on the line \(x+9y+c=0\), then \(c\) is

• A. 12
• B. 14
• C. 13
• D. 15

Answer 1

The Correct Option is B

To find the value of \( c \) when the other two vertices of the parallelogram lie on the line \( x+9y+c=0 \), we start by noting that the midpoints of the diagonals of a parallelogram coincide. Let the unknown vertices be \((x_1, y_1)\) and \((x_2, y_2)\). The known vertices are \((2, 1)\) and \((-3, -4)\). The midpoint of diagonal connecting \((2, 1)\) and \((-3, -4)\) is:

\[\left(\frac{2 + (-3)}{2}, \frac{1 + (-4)}{2}\right) = \left(-\frac{1}{2}, -\frac{3}{2}\right)\]

For the other diagonal, connecting \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint should be the same:

\[\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(-\frac{1}{2}, -\frac{3}{2}\right)\]

Equating the midpoints gives the equations:

\[x_1 + x_2 = -1\]

\[y_1 + y_2 = -3\]

Since \((x_1, y_1)\) and \((x_2, y_2)\) lie on the line \(x + 9y + c = 0\), we have:

\[x_1 + 9y_1 + c = 0\]

\[x_2 + 9y_2 + c = 0\]

Adding these two equations, we get:

\[(x_1 + x_2) + 9(y_1 + y_2) + 2c = 0\]

Substituting the sums we calculated:

\[-1 + 9(-3) + 2c = 0\]

Simplifying yields:

\[-1 - 27 + 2c = 0\]

\[2c = 28\]

\[c = 14\]

Thus, the value of \( c \) is \( 14 \).

Answer 2

Concept: In a parallelogram, the diagonals bisect each other. That means the midpoints of both diagonals are the same.

Given points on the diagonals are: \( (2, 1) \) and \( (-3, -4) \) 

Midpoint of these two points is: \[ \left( \frac{2 + (-3)}{2}, \frac{1 + (-4)}{2} \right) = \left( \frac{-1}{2}, \frac{-3}{2} \right) \]

We are given that this midpoint lies on the line: \( x + 9y + c = 0 \)

Substitute \( x = -\frac{1}{2} \) and \( y = -\frac{3}{2} \) into the equation:

\[ -\frac{1}{2} + 9 \cdot \left( -\frac{3}{2} \right) + c = 0 \]

\[ -\frac{1}{2} - \frac{27}{2} + c = 0 \]

\[ \Rightarrow -\frac{28}{2} + c = 0 \Rightarrow -14 + c = 0 \]

\[ \Rightarrow c = 14 \]

∴ Correct answer is (B): \( \boxed{14} \)